22edo
22edo, or 22 equal divisions of the octave, is the equal tuning with a step size of 1200/22 ~= 54.5 cents, dividing 2/1 into 22 steps.
22edo is the fourth-smallest EDO with a diatonic (5L 2s) MOS scale formed by a chain of fifths, which has a hardness of 4:1. It achieves this with a perfect fifth tuned sharpward (~709¢) so that 9/8 and 8/7 are the same interval. Its logic is therefore that of Archy (or Superpyth) temperament, rather than Meantone: that is, the minor and major thirds available in the diatonic MOS approximate the septal thirds, 7/6 and 9/7, often called "subminor" and "supermajor" (including in the ADIN system for melodic qualities, which will be used in the remainder of this article).
As an even EDO, 22edo includes the 600¢ tritone familiar from 12edo, but it divides neither the perfect fourth nor fifth in half, meaning that it does not include semifourths or neutral thirds. It divides the perfect fourth (9\22) in three, however, implying that a tetrachord of three equal intervals is possible in 22edo. 22edo also includes 11edo as a subset, and similarly to 6edo (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.
22edo distinguishes its native subminor and supermajor thirds from approximations to 5-limit intervals, 6/5 and 5/4, which ADIN calls "nearminor" and "nearmajor" thirds. As a result, 22 is perhaps the smallest EDO that can be considered to include full 7-limit harmony, as it is the first to distinctly (and consistently) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the 11th harmonic, although many 11-limit intervals are not distinguished from 5-limit intervals (e.g. 11/9 is mapped to the same interval as 6/5), as well as the 17th.
General theory
Edostep interpretations
22edo's edostep has the following interpretations in the 7-limit:
- 25/24 (the difference between 5/4 and 6/5)
- 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)
- 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
- 49/48 (the difference between 8/7 and 7/6)
- 81/80 (the difference between 10/9 and 9/8)
Including prime 11, it additionally serves as:
- 22/21 (the difference between 7/6 and 11/9, or 14/11 and 4/3)
- 33/32 (the difference between 4/3 and 11/8, or 12/11 and 9/8)
- 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)
Tempered commas
Important commas tempered out by the 11-limit of 22et include:
- 50/49 (jubilismic), equating 7/5 and 10/7 to exactly half of an octave.
- 55/54 (telepath), equating 6/5 with 11/9
- 64/63 (archytas), equating 16/9 with 7/4
- 99/98 (mothwellsmic), equating 14/11 with 9/7
- 245/243 (sensamagic), equating a stack of two 9/7s to 5/3
- 250/243 (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)
Regular temperaments associated with these are discussed in #Temperaments and generators. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form 8/5 (orwellismic). In terms of S-expressions, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.
JI approximation
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that 25/24, the chroma separating the classical major triad 4:5:6 and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, 49/48, the chroma separating 6:7:8 from its complement, is exaggerated, in fact to the same size as 25/24. This gives 7/5 the most damage out of the 7-odd-limit, tuning it (and thus 10/7) to the semioctave at 600¢. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3¢ sharp.
22edo also approximates the interval 11/10 to within 1.4¢, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic.
Among the higher primes, 22edo approximates 17/16 as two steps and 32/29 as three steps, and one step of 22edo is extremely close to 32/31. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as 29/24, which is only about 0.35¢ off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +7.1 | -4.5 | +13.0 | -5.9 | -22.3 | +4.1 | -24.8 | +26.3 | +6.8 | +0.4 |
| Relative (%) | 0.0 | +13.1 | -8.2 | +23.8 | -10.7 | -41.0 | +7.6 | -45.4 | +48.2 | +12.4 | +0.8 | |
| Steps
(reduced) |
22
(0) |
35
(13) |
51
(7) |
62
(18) |
76
(10) |
81
(15) |
90
(2) |
93
(5) |
100
(12) |
107
(19) |
109
(21) | |
Intervals and notation
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff).
The "native-fifths" system mentioned here is the most commonly used system, and the one that most microtonal notation systems support by default. It is derived through stacking 22edo's tempered version of 3/2 and assigning names accordingly. As a result, the nominals (C, D, E, F, G, ...) follow the diatonic MOS, where the distances between the large steps (C-D, D-E, F-G, G-A, A-B) are 4 EDO steps, and the distances between the small steps (E-F, B-C) are 1 EDO step. Therefore, a sharp corresponds to +3 EDO steps while a flat corresponds to -3 (representing the diatonic chroma in each case). Each sharp or flat can be split into three distinct notes, so we use the accidental ^ to raise by one EDO step and v to lower by one EDO step.
In the other proposed notation systems aside from native fifths, a sharp corresponds to +1 EDO step, while a flat corresponds to -1. The Zarlino notation here uses the ternary Zarlino scale (see #Zarlino diatonic), or Ptolemy's intense diatonic, as its basic scale (which prioritizes the 5-limit, whereas native fifths prioritize 2.3.7), and uses to its advantage the fact that one step of 22edo maps to both 25/24 and 81/80 (a property of Porcupine temperament). Pajara uses the 10-note Pajara scale (see #Pajara) as its basis, which is generated by a perfect fifth but splits the octave in two to reach intervals of the full 7-limit relatively easily; the scale has four 2-step and one 3-step intervals per half-octave, which differ in size by a diatonic minor second, which is 1 step in 22edo.
The ADIN system uses the labels "nearminor" and "nearmajor" for intervals that may otherwise be called "classic(al)", "pental", or "ptolemaic" minor/major, which are terms used to describe the simple 5-limit intervals to which they correspond. Note also that as "subminor" and "supermajor" intervals are the minor and major intervals of the diatonic MOS in 22edo, unqualified "major" and "minor" by default refer to these.
JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
| Edostep | Cents | 11-limit add-17 approximation |
Notation | Interval category (ADIN) | ||
|---|---|---|---|---|---|---|
| Native-fifths (ups & downs) |
Blackdye/Zarlino (Vector) |
Pajara decatonic | ||||
| 0 | 0 | 1/1 | C | C | 0 | Perfect unison |
| 1 | 55 | 25/24, 28/27, [33/32], 36/35 | ^C, Db | C# | 1b | (Sub)minor second |
| 2 | 109 | [16/15], 15/14, 18/17, [17/16] | vC#, ^Db | Db | 1 | Nearminor second |
| 3 | 164 | 10/9, [11/10], 12/11 | C#, vD | D | 1# | Nearmajor second |
| 4 | 218 | 8/7, 9/8, 17/15 | D | D# | 2 | (Super)major second |
| 5 | 273 | 7/6 | ^D, Eb | Ebb / Dx | 2# | (Sub)minor third |
| 6 | 327 | 6/5, 11/9 | vD#, ^Eb | Eb | 3b | Nearminor third |
| 7 | 382 | [5/4] | D#, vE | E | 3 | Nearmajor third |
| 8 | 436 | [9/7], 14/11, 32/25 | E | E# | 4b | (Super)major third |
| 9 | 491 | 4/3 | F | F | 4 | Perfect fourth |
| 10 | 545 | 11/8, 15/11 | ^F, Gb | F# | 4# | Near fourth |
| 11 | 600 | 7/5, 10/7, [17/12] | vF#, ^Gb | Gbb / Fx | 5 | Tritone |
| 12 | 655 | 16/11 | F#, vG | Gb | 6b | Near fifth |
| 13 | 709 | 3/2 | G | G | 6 | Perfect fifth |
| 14 | 764 | [14/9], 11/7, 25/16 | Ab | G# | 6# | (Sub)minor sixth |
| 15 | 818 | [8/5] | vG#, ^Ab | Ab | 7 | Nearminor sixth |
| 16 | 873 | 5/3, 18/11 | G#, vA | A | 7# | Nearmajor sixth |
| 17 | 927 | 12/7 | A | A# | 8b | (Super)major sixth |
| 18 | 982 | 7/4, 16/9, 30/17 | ^A, Bb | Bbb / Ax | 8 | (Sub)minor seventh |
| 19 | 1036 | 9/5, [20/11], 11/6 | vA#, ^Bb | Bb | 9b | Nearminor seventh |
| 20 | 1091 | [15/8], 28/15, 17/9, [32/17] | A#, vB | B | 9 | Nearmajor seventh |
| 21 | 1145 | 48/25, 27/14, [64/33], 35/18 | B | Cb | 9# | (Super)major seventh |
| 22 | 1200 | 2/1 | C | C | 10 | Octave |
Structural theory
Temperaments and generators
Whole-octave temperaments
22edo has five distinct intervals that generate octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).
3\22 serves as 10/9, 11/10, and 12/11 simultaneously. The temperament associated with this equivalence is Porcupine, where the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like 15 and 37, as well as 29edo aside from the mapping of 7.
5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in Orwell temperament, while if stacked further, four 7/6s are made to reach 15/8, so that 3/1 is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like 31 and 53edo, though note that the 11-limit is less accurate than the 7-limit component in general.
7\22 represents a flattened 5/4, five of which stack to 3/1, which is Magic temperament. The deficit between the octave and three 5/4s, 128/125, is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like 19 and 41edo.
Finally, 9\22 represents 4/3, two of which stack to 7/4 in Archy/Superpyth temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in 27 and 49edo; this is not shared with even sharper tunings of Archy, such as 37edo.
Split-octave temperaments
22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is Pajara, generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings 10edo and 12edo). This general structure without prime 7, known as Diaschismic, however, is supported by notable EDOs such as 34 and 46edo.
Temperaments of 11edo
Important temperaments that 22edo borrows from 11edo include Orgone (a 2.7.11 structure generated by the nearminor third, so that two of them form 16/11 and three form 7/4; note that this is in fact every other step of Porcupine), as well as Sentry (where two 9/7s reach 5/3, and in this case serves as every other step of Magic).
Tertian structure
22edo has four clear qualities of "thirds" that can serve as mediants in a chord bounded by a fifth. These are the subminor (273¢, 5\22), nearminor (327¢, 6\22), nearmajor (382¢, 7\22), and supermajor (436¢, 8\22) thirds, which reflect the intervals 7/6, 6/5, 5/4, and 9/7 respectively. As the gap between 6/5 and 5/4 is the same as that between 7/6 and 6/5 (or 5/4 and 9/7), 22edo's tertian structure is keemic.
| Quality | Subminor | Nearminor | Nearmajor | Supermajor |
|---|---|---|---|---|
| Cents | 273 | 327 | 382 | 436 |
| Just interpretation | 7/6 (+5.9¢) | 6/5 (+11.6¢) | 5/4 (-4.5¢) | 9/7 (+1.3¢) |
Diatonic thirds are bolded.
Scales
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).
One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.
Pythagorean diatonic
This is the diatonic scale most directly analogous in structure to the 12edo diatonic, given its MOS form. Advantages of using it include the fact that all steps are what they appear to be - for example, D-G and G-C are both perfect fifths - and that it appears as a subset of the chain of fifths itself. One key difference is that the major and minor thirds do not get mapped to the expected 5-limit interpretations, but rather to the supermajor and subminor thirds of 22edo. This is good for septimal harmony, not so much for 5-limit harmony (to get 5-limit thirds, you have to go 9 steps up!). A downside, or more generally a significant awkwardness, to using this system is the fact that due to the minor second being so small, the chromatic semitone is massive - closer to a whole tone than a proper semitone (in fact, it is enharmonic to the small whole-tone used in the Zarlino system). This can make interval classifications somewhat unintuitive if ups and downs are not used - for instance, a Nearmajor triad, 0-7-13, is C-D#-G (meaning that a third-sized interval is technically a second) - and in fact 22edo is the first edo other than 15edo (which is strange in its own right) to require ups and downs to notate all intervals without problems with intuition like this (Edos like 17 and 10 utilize semisharps and semiflats, which 22edo cannot use as it does not have neutral intervals.)
Zarlino diatonic
A faithful way of representing the 5-limit diatonic structure in 22edo (which, unlike its just counterpart, keeps the 7-limit convenient to access) is to use 5/4 as the scale's major third, and likewise 5/3 and 15/8 as the major sixth and seventh respectively. The scale pattern for zarlino is 4-3-2-4-3-4-2. This encounters problems with existing familiarity with notation (for example, one of D-G and G-C must now be a flat 5th, called a "wolf fifth" and representing 16/11 as opposed to 3/2), however it does have one thing going for it: in 22edo in particular (and a family of edos including 15, 29, and 37), the interval which separates Pythagorean intervals from their 5-limit counterparts is the exact same as the interval separating 5-limit major and minor intervals. This means that, if # and b are used to represent this interval as an accidental, no additional accidentals are necessary. This is why the 3-step interval is a "Nearmajor second" along with being a chromatic semitone.
Pajara
Note that the 1-step interval which serves as the 3-limit diatonic semitone is the very same as the 5-limit chromatic semitone, and that the 3-step interval serving as the 3-limit chromatic semitone also happens to map to a certain kind of large 5-limit diatonic semitone (such that it and the chromatic semitone stack to the 4-step whole tone). This suggests that if there was a way to swap the diatonic and chromatic semitones, we could faithfully represent the full 7-limit, including 5.
And as it turns out, there is a way to do that. We start with the pentatonic scale, not only because it's closer to even but because the interval between its small and large steps is exactly the 1-step interval we want to function as our chroma. Then, we add another pentatonic scale that is offset by a tritone from the first one. This results in the decatonic "Pajara[10]" scale (3-2-2-2-2-3-2-2-2-2), where every note has a corresponding note a tritone apart. It solves the problem of representing intervals of both 5 and 7 by introducing three new ordinal classes to provide space for 7-limit intervals to fit on their own degrees of the scale. That way, 7/4 isn't a subminor seventh, it's a major version of the Pajara 8-step. One can even define a notation system for Pajara, wherein the notes are numbered 0-9 and # and b represent alterations by a single step. Pajara retains the property where most notes have a fifth over them.
To extend to the 11-limit, Pajara[12] (1-2-2-2-2-2-1-2-2-2-2-2) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step.
Symmetric scale
One possible scale of 22edo, as mentioned previously, is the Pajara[10] decatonic scale, represented as ├─┴─┴──┴─┴─┴─┴─┴──┴─┴─┤. This scale can be explored here. Below is a chart of its five modes, ordered by rotation. Some names are from Paul Erlich.
| Chart | 2 | 3 | 4 | 6 | 8 | 9 | Mode on fifth | Mode on fourth | |
|---|---|---|---|---|---|---|---|---|---|
| Dynamic minor | ├─┴─┴─┴─┴──┴─┴─┴─┴─┴──┤ | minor | minor | dim | perfect | minor | minor | ||
| Static minor | ├─┴─┴─┴──┴─┴─┴─┴─┴──┴─┤ | minor | minor | perfect | perfect | minor | major | ||
| Static major | ├─┴─┴──┴─┴─┴─┴─┴──┴─┴─┤ | minor | major | perfect | perfect | major | major | ||
| Dynamic major | ├─┴──┴─┴─┴─┴─┴──┴─┴─┴─┤ | major | major | perfect | perfect | major | major | ||
| Augmented | ├──┴─┴─┴─┴─┴──┴─┴─┴─┴─┤ | major | major | perfect | aug | major | major |
Pentachordal scale
This scale is constructed from two identical "pentachords" and the semioctave, and is represented as ├─┴─┴─┴─┴─┴──┴─┴─┴─┴──┤. Unlike the symmetric scale, there are ten distinct modes of the pentachordal scale. Ordered by rotation, they are as follows:
| Chart | 2 | 3 | 4 | 6 | 8 | 9 | Mode on fifth | Mode on fourth | |
|---|---|---|---|---|---|---|---|---|---|
| Bediyic | ├─┴─┴─┴─┴─┴──┴─┴─┴─┴──┤ | minor | minor | dim | perfect | minor | minor | Hininic | - |
| Alternate minor (Skoronic) | ├─┴─┴─┴─┴──┴─┴─┴─┴──┴─┤ | minor | minor | dim | perfect | minor | major | Aujalic | - |
| Moriolic | ├─┴─┴─┴──┴─┴─┴─┴──┴─┴─┤ | minor | minor | perfect | perfect | major | major | Mielauic | Hininic |
| Standard major (Staimosic) | ├─┴─┴──┴─┴─┴─┴──┴─┴─┴─┤ | minor | major | perfect | perfect | major | major | Prathuic | Aujalic |
| Sebaic | ├─┴──┴─┴─┴─┴──┴─┴─┴─┴─┤ | major | major | perfect | aug | major | major | - | Mielauic |
| Awanic | ├──┴─┴─┴─┴──┴─┴─┴─┴─┴─┤ | major | major | perfect | aug | major | major | - | Prathuic |
| Standard minor (Hininic) | ├─┴─┴─┴──┴─┴─┴─┴─┴─┴──┤ | minor | minor | perfect | perfect | minor | minor | Moriolic | Bediyic |
| Aujalic | ├─┴─┴──┴─┴─┴─┴─┴─┴──┴─┤ | minor | major | perfect | perfect | minor | major | Staimosic | Skoronic |
| Alternate major (Kielauic) | ├─┴──┴─┴─┴─┴─┴─┴──┴─┴─┤ | major | major | perfect | perfect | major | major | Sebaic | Moriolic |
| Prathuic | ├──┴─┴─┴─┴─┴─┴──┴─┴─┴─┤ | major | major | perfect | perfect | major | major | Awanic | Staimosic |
Some names are from Paul Erlich.
Blackdye
Blackdye is a rank-3 scale similar to zarlino diatonic, which attempts to compromise between zarlino and Pythagorean diatonic in a way, by including a few intervals from both at once. Blackdye can be thought of as dividing each 4-step whole tone into a 3-step tone and a single step called an aberrisma, which separates zarlino and Pythagorean intervals. The blackdye scale pattern is 3-1-3-2-3-1-3-1-3-2. One way to use blackdye is to essentially treat it as multiple overlapping diatonics, which one can modulate between.
Other scales
| Name | Chart | Notes |
|---|---|---|
| Onyx | ├──┴──┴──┴───┴──┴──┴──┤ | Approximate Greek scale (equable diatonic), basic MOS of Porcupine. |
| Gramitonic (4L5s) | ├──┴─┴──┴─┴──┴─┴──┴─┴─┤ | Basic MOS of Orwell temperament. |
| Zarlino diatonic | ├─┴───┴──┴───┴─┴───┴──┤ | Greek scale (intense diatonic). Zarlino rank-3 diatonic. |
| Mosdiatonic | ├┴───┴───┴───┴┴───┴───┤ | Greek scale (Pythagorean or Archytas diatonic). Basic MOS of Superpyth. |
| Zarlino pentatonic | ├─────┴──┴───┴─────┴──┤ | One possible pentatonic analog to the Zarlino diatonic. |
| Pentic | ├────┴───┴───┴────┴───┤ | Basic MOS of Superpyth |
Triads and tetrads
Triads bounded by P5
| Name | 1 | 2 | Bounding interval | Edostep | Chart |
|---|---|---|---|---|---|
| Sus4 triad | Perfect 4th | Supermajor 2nd | Perfect 5th | [0 9 13] | ├────────┴───┴────────┐ |
| Supermajor triad | Supermajor 3rd | Subminor 3rd | Perfect 5th | [0 8 13] | ├───────┴────┴────────┐ |
| Nearmajor triad | Nearmajor 3rd | Nearminor 3rd | Perfect 5th | [0 7 13] | ├──────┴─────┴────────┐ |
| Nearminor triad | Nearminor 3rd | Nearmajor 3rd | Perfect 5th | [0 6 13] | ├─────┴──────┴────────┐ |
| Subminor triad | Subminor 3rd | Supermajor 3rd | Perfect 5th | [0 5 13] | ├────┴───────┴────────┐ |
| Sus2 triad | Supermajor 2nd | Perfect 4th | Perfect 5th | [0 4 13] | ├───┴────────┴────────┐ |
Tetrads with P5th
Harmonic tetrads
These represent the conventional structure of a 4:5:6:7 harmonic tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect median above the middle note. This is the characteristic kind of chord found in Pajara temperament, and can also be thought of the stacking of a fifth-bounded triad and a fourth-bounded triad.
| Name | 1 | 2 | 3 | 4 | Bounding interval 1 | Bounding interval 2 | Bounding interval 3 | Edostep | Chart |
|---|---|---|---|---|---|---|---|---|---|
| Nearmajor harmonic tetrad | Nearmajor 3rd | Nearminor 3rd | Subminor 3rd | Supermajor 2nd | Perfect 5th | Subminor 7th | Perfect 8ve | [0 7 13 18] | ├──────┴─────┴────┴───┤ |
| Nearminor harmonic tetrad | Nearminor 3rd | Nearmajor 3rd | Supermajor 2nd | Subminor 3rd | Perfect 5th | Supermajor 6th | Perfect 8ve | [0 6 13 17] | ├─────┴──────┴───┴────┤ |
Diatonic tetrads
These represent the conventional structure of an 8:10:12:15 diatonic major tetrad, but generalized such that any middle interval (and thus any top interval) can be used. They take the form of a fifth-bounded triad, with an additional note a perfect fifth above the middle note. The sus4 diatonic tetrad is the same as the triad version up to inversion.
| Name | 1 | 2 | 3 | Bounding interval 1 | Bounding interval 2 | Edostep | Chart |
|---|---|---|---|---|---|---|---|
| Supermajor diatonic tetrad | Supermajor 3rd | Subminor 3rd | Supermajor 3rd | Perfect 5th | Supermajor 7th | [0 8 13 21] | ├───────┴────┴───────┴┤ |
| Nearmajor diatonic tetrad | Nearmajor 3rd | Nearminor 3rd | Nearmajor 3rd | Perfect 5th | Nearmajor 7th | [0 7 13 20] | ├──────┴─────┴──────┴─┤ |
| Nearminor diatonic tetrad | Nearminor 3rd | Nearmajor 3rd | Nearminor 3rd | Perfect 5th | Nearminor 7th | [0 6 13 19] | ├─────┴──────┴─────┴──┤ |
| Subminor diatonic tetrad | Subminor 3rd | Supermajor 3rd | Subminor 3rd | Perfect 5th | Subminor 7th | [0 5 13 18] | ├────┴───────┴────┴───┤ |
| Sus2 diatonic tetrad | Supermajor 2nd | Perfect 4th | Supermajor 2nd | Perfect 5th | Supermajor 6th | [0 4 13 17] | ├───┴────────┴───┴────┤ |
